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Approaching a vertex in a shrinking domain under a nonlinear flow

Authors:	M A Herrero M Ughi and J J L Velázquez

Institution:	(1) Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain;(2) Dipartimento di Scienze Matematiche, Facoltà di Ingegneria, Università di Trieste, 34100 Trieste, Italia;(3) Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain

Abstract:	We consider here the homogeneous Dirichlet problem for the equation $u_t = u\Delta u - \gamma \|\nabla u\|^2 \quad \mathrm{with} \quad \gamma \in R, u \geq 0$ , in a noncylindrical domain in space-time given by $\|x\| \leq R(t) = (T - t)^p, \quad \mathrm{with} \quad p > 0$ . By means of matched asymptotic expansion techniques we describe the asymptotics of the maximal solution approaching the vertex x=0, t=T, in the three different cases p>1/2, p=1/2(vertex regular), p<1/2 (vertex irregular).

Keywords:	35B40 35K60 35k65
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